A blob of plasma is ejected from an accreting black hole's jet with speed at an angle…

PHYSICS · Subjective · Fermi University Physics · Relativity & Gravitation · Astrophysics & Cosmology · Special relativity: spacetime and kinematics

Stars from the Inside, Ch Accretion power: mass transfer, discs and the weighing of black holes

Problem

A blob of plasma is ejected from an accreting black hole's jet with speed βc\beta c at an angle θ\theta to the line of sight, measured in the frame of the binary. Special relativity applies; there is no gravitational light bending and the binary is at rest relative to the observer. (a) By comparing the light-travel time from two points on the blob's path a short time Δt\Delta t apart, measured in the binary's rest frame, derive the apparent transverse speed on the sky, βapp=βsin⁡θ1−βcos⁡θ.\beta_{\rm app}=\frac{\beta\sin\theta}{1-\beta\cos\theta}. (b) Show that, at fixed β\beta, βapp\beta_{\rm app} is greatest when cos⁡θ=β\cos\theta=\beta, and that the maximum value is γβ\gamma\beta, with γ=(1−β2)−1/2\gamma=(1-\beta^2)^{-1/2}. (c) A jet blob has β=0.98\beta = 0.98. Find the angle at which its apparent speed is greatest, and the value of that maximum apparent speed.

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